If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-Viro construction for two Morita equivalent fusion categories gives the same 3d TQFT. Equivalently said, if $\mathcal{C}$ has Froboenius algebra objects $A$ we can contruct Morita equivalent categories by condensing $A$, which turns out the be the category of bimodules of $A$, and this corresponds to all possible Lagrangian algebra objects of the Drinfeld center.
My question is whether this extends to fusion 2-cateogries and 4d TQFTs. I know that there are state-sum constructions of 4d TQFTs from the datum of a braided fusion 1-category (which can be understood as a fusion 2-category with one simple object), like the Crane-Yetter state-sum, or more generally from the datum of a spherical fusion 2-category (by Douglas and Reutter). It seems that still the braided fusion 2-category associated to these 4d TQFT is the Drinfeld center of the fusion 2-category we started with (correct me if I am wrong), and this is an invariant under Morita equivalences (for instance https://arxiv.org/abs/2211.04917). However in the case of fusion 1-cateogries the Drinfeld center is, in a certain sense, the unique invariant of the Morita equivalence classes, since for any Froboenius algebras of $\mathcal{C}$ there is a Lagrangian algebra object of $Z(\mathcal{C})$. It is not clear to me whether this is true also in the case of fusion 2-cateogries. Actually it seems to me that the Drinfeld center of a fusion 2-cateogory is not the only invariants of its Morita equivalence class.
One of the origins of my confusion is the following. The Crane-Yetter state sum for a modular tensor category produces a 4d TQFT which is an invertible theory, and can be thought as the anomaly inflow theory for the 3d Reshetikhin-Turaev theory of the starting MTC. However if the MTC has some commutative Froboenius algebra objects we could condense them producing Morita equivalent categories, and I would expect the Drinfeld center to have a corresponding Lagrangian algebra object. But this cannot be the case if the 4d TQFT is invertible... In other words all the MTCs have the same Drinfeld center (it is trivial) but not all of them are Morita equivalent. Is there something beyond the Drinfeld center which differentiate the various Morita equivalence classes?
